Methods and formulas for the goodness-of-fit tests for Fit Cox Model in a Counting Process Form

The degrees of freedom for the goodness-of-fit tests are the sum of the degrees of freedom for the terms in the model. This sum equals the number of parameters in the model.

The calculation of the chi-square statistic depends on the test. When the response variable has no tied response times, then the score test is identical to the well-known log-rank test.

Under the null hypothesis, the test statistic for each type of test has an asymptotic chi-square distribution. The asymptotic distribution is valid when the number of observed events is large compared to the number of parameters in the model. For categorical predictors, the number of events in each level must also be large enough.

Likelihood ratio test

For the likelihood ratio test, the test statistic has the following form:

where is the appropriate model partial log-likelihood function.

Wald test

For the Wald test, the test statistic has the following form:

where is the Fisher information matrix.

If the design has clusters, the calculations make use of the robust variance from Lin & Wei (1989)¹. Let be the matrix of score residuals. Then, the robust variance covariance matrix has the following form:

where and is the collapsed score residual matrix. To obtain the collapsed score residual matrix, replace each cluster of score residual rows by the sum of those residual rows.

Then, the Wald test statistic has the following form:

Score test

For the score test, the test statistic has the following form:

where

and

If the design has clusters, the test statistic has the following modification:

where is the collapsed score residual matrix for . To obtain the collapsed score residual matrix, replace each cluster of score residual rows by the sum of those residual rows.

The p-value has the form:

where is a random variable that follows a chi-square distribution with degrees of freedom. is the test statistic.